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\cl {\bf PARADOXI}
\noi
\cl {RUSSELL'S PARADOX}
\noi
Let $T$ denote the set of all sets that don't contain themselves.
Symbolically, we write $T=\{x:x\notin x\}$.
For example, the set $A_1 =\{all \ forks\}$ is not a fork and thus
does not contain itself, so 
$A_1 \notin A_1$ and thus
$A_1 \in T$. On the other hand, the 
set $A_2 =\{all \ things \ that \ aren't \ forks\}$ is not a fork
and  thus contains itself. That is, $A_2 \in A_2 $ whence
$A_2 \notin T$.
\noi
Question: Does $T$ contain itself or not? 
\noi
1) Suppose so. Thus $T\in T$. But the defining property of $T$ is 
that it consists {\it only} of those sets that don't contain 
themselves. So since $T \in T$, then by this defining property
we must have that $T$ doesn't contain itself. Hence $T\notin T$.
In other words:
$$T\in T \Longrightarrow T\notin T$$
\noi
2) Suppose not. Thus $T \notin T$  and $T$ is a set that does
not contain itself. But the defining property of $T$ is that it
consists of {\it all} of the sets that don't contain themselves.
And since $T\notin T$, $T$ is a set that doesn't contain itself 
and thus belongs in the set $T$ (otherwise $T$ is not the set
of {\it all} sets that don't contain themselves). Thus $T\in T$
and 
we have that
$$T\notin T \Longrightarrow T\in T$$
\noi

\cl {THE BARBER'S PARADOX}
\noi
In a small town a male barber shaves precisely all of those men in
the town that do not shave themselves. 
\noi
Question: Does the barber shave himself?
\noi
If so, then since, by virtue of being the barber, he shaves only 
those men that do not shave themselves, the fact that he 
shaves himself implies that he is one of those that do not shave 
themselves.
\noi
If not, then (besides being hairy) he is one of the men who don't
shave themselves and thus he must shave himself. 
\noi
\cl {LIAR'S PARADOX} 
\noi
1) This statement is false. 
\noi
or
\noi
2) The next statement  is false. The previous statement is
true
\noi

\cl{BERRY PARADOX}
\noi
Consider the Berry sentence:
\noi
{\it The smallest positive integer not nameable in under eleven words.}
\noi
For example, the number 
$1,273,461$
can be described by the 11 words
{\it One million two hundred  seventy three thousand
four hundred sixty one}.
$1,273,461$ can also be described by the 12 words (
{\it A one followed by a two, seven, three, four, six, and one})
\noi
{\it Berry Sentence: The smallest positive integer not nameable in under eleven words.}
\noi
It is reasonable to assume that the Berry sentence specifies  a number explicitly: after all, there are a finite number of sentences of less than eleven words, and some finite subset of them specify unique positive integers, so there is clearly some positive number that is the smallest integer not in that finite set.
But the Berry sentence itself is a specification for that number in only ten words!

\cl{ HILBERT'S HOTEL}
\noi
Imagine a hotel with a finite number of rooms, and assume that all the rooms are occupied. A new guest arrives and asks for a room. "Sorry" - says the proprietor - "but all the rooms are occupied." 
\noi
 Now let us imagine a hotel with an infinite number of rooms, and all the rooms are occupied. To this hotel, too, comes a new guest and asks for a room. "But of course!" - exclaims the proprietor, and he moves the person previously occupying room Room1 into Room2, the person from Room2 into Room3, the person from Room3 into Room4, and so on... And the new customer receives Room1, which becomes free as a result of these transpositions.
\noi
Let us imagine now a hotel with an infinite number of rooms, all taken up, and an {\bf infinite} number of new guests who come in, and ask for rooms.
"Certainly, " says the proprietor, "just wait a minute." He moves the occupant of Room1 into Room2, the occupant of Room2 into Room4, the occupant of Room3 into Room6, and so on, and so on...
\noi
Now all odd numbered rooms become free and the infinity of new guests can easily be accommodated in them. 

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%Question: Are the above statements true or false?